Lecture 8: Finance: GBM model for share prices, futures and options, hedging, and the Black-Scholes equation Outline: GBM as a model of share prices futures and options, hedging derivation of the Black-Scholes equation

Geometric Brownian motion as a model of share prices Empirical facts:

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate:

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate:

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate: Stock return (δs/s) fluctuations have a very short correlation time

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate: Stock return (δs/s) fluctuations have a very short correlation time

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate: Stock return (δs/s) fluctuations have a very short correlation time This is GBM (that we already studied).

Geometric Brownian motion as a model of share prices Empirical facts: On average, stocks appreciate in value at an approximately constant rate: Stock return (δs/s) fluctuations have a very short correlation time This is GBM (that we already studied). σ is called the volatility.

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future.

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option)

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option).

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.)

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.) Options are insurance.

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.) Options are insurance. A derivative has a price. Our aim here is to determine the fair price.

derivative securities Future: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future. Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares at a particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.) Options are insurance. A derivative has a price. Our aim here is to determine the fair price. Since a derivative security has a price, one can also buy or sell a future or option on it (derivative of a derivative).

Fair price of a future(?) What is the fair price of a future (per share of a stock S )?

Fair price of a future(?) What is the fair price of a future (per share of a stock S )? A possible answer: S will fluctuate according to GBM. We have already calculated the distribution of prices at T :

Fair price of a future(?) What is the fair price of a future (per share of a stock S )? A possible answer: S will fluctuate according to GBM. We have already calculated the distribution of prices at T :

Fair price of a future(?) What is the fair price of a future (per share of a stock S )? A possible answer: S will fluctuate according to GBM. We have already calculated the distribution of prices at T : The average value of S at time T is

Fair price of a future(?) What is the fair price of a future (per share of a stock S )? A possible answer: S will fluctuate according to GBM. We have already calculated the distribution of prices at T : The average value of S at time T is Shouldn’t this be the fair price of the future?

Hedging No!

Hedging No! This argument ignores the possibility of hedging.

Hedging No! This argument ignores the possibility of hedging. The seller could just buy a share of S herself at t = 0 and sell it to the buyer at maturity ( T ). This is riskless for the seller. Cost to the seller: She had to spend some of her own money ( =S 0 ) on that share at t = 0. That money could (risklessly) have been invested in T-bills, which give a rate r 0. By buying the share, she has lost the chance to see her original cash become S 0 exp(r 0 T).

Hedging No! This argument ignores the possibility of hedging. The seller could just buy a share of S herself at t = 0 and sell it to the buyer at maturity ( T ). This is riskless for the seller. Cost to the seller: She had to spend some of her own money ( =S 0 ) on that share at t = 0. That money could (risklessly) have been invested in T-bills, which give a rate r 0. By buying the share, she has lost the chance to see her original cash become S 0 exp(r 0 T). Therefore, the fair price of the future contract is S 0 exp(r 0 T)

Hedging No! This argument ignores the possibility of hedging. The seller could just buy a share of S herself at t = 0 and sell it to the buyer at maturity ( T ). This is riskless for the seller. Cost to the seller: She had to spend some of her own money ( =S 0 ) on that share at t = 0. That money could (risklessly) have been invested in T-bills, which give a rate r 0. By buying the share, she has lost the chance to see her original cash become S 0 exp(r 0 T). Therefore, the fair price of the future contract is S 0 exp(r 0 T) (Any other price would allow arbitrage).

Options: call European call option: What is the fair price to charge for the option to buy a share at a strike price X at time T ?

Options: call European call option: What is the fair price to charge for the option to buy a share at a strike price X at time T ? If S(T) > X, the buyer will certainly take advantage of the option. (He could immediately resell it and gain the difference.)

Options: call European call option: What is the fair price to charge for the option to buy a share at a strike price X at time T ? If S(T) > X, the buyer will certainly take advantage of the option. (He could immediately resell it and gain the difference.) If S(T) < X, the buyer will not exercise the option (he can get it for less than X on the market).

Options: call European call option: What is the fair price to charge for the option to buy a share at a strike price X at time T ? If S(T) > X, the buyer will certainly take advantage of the option. (He could immediately resell it and gain the difference.) If S(T) < X, the buyer will not exercise the option (he can get it for less than X on the market). This means a net cost to the seller of

Options: put European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T ?

Options: put European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T ? If S(T) < X, the buyer of the option will certainly take advantage.

Options: put European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T ? If S(T) < X, the buyer of the option will certainly take advantage. (He could then immediately buy it on the market instead and gain the difference.)

Options: put European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T ? If S(T) < X, the buyer of the option will certainly take advantage. (He could then immediately buy it on the market instead and gain the difference.) If S(T) > X, the buyer will not exercise the option (he can sell it for more than X on the market).

Options: put European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T ? If S(T) < X, the buyer of the option will certainly take advantage. (He could then immediately buy it on the market instead and gain the difference.) If S(T) > X, the buyer will not exercise the option (he can sell it for more than X on the market). This means a net cost to the seller of

The strategy to find the right price at no risk to the seller (European call option) Let f(t) be the value of the option at t, 0 < t ≤ T.

The strategy to find the right price at no risk to the seller (European call option) Let f(t) be the value of the option at t, 0 < t ≤ T. If we were already at time T (now we know S(T) ), the fair price is just.

The strategy to find the right price at no risk to the seller (European call option) Let f(t) be the value of the option at t, 0 < t ≤ T. If we were already at time T (now we know S(T) ), the fair price is just. Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer.

The strategy to find the right price at no risk to the seller (European call option) Let f(t) be the value of the option at t, 0 < t ≤ T. If we were already at time T (now we know S(T) ), the fair price is just. Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer. If, on the other hand, if S(T - dt) < X, she need do nothing.

The strategy to find the right price at no risk to the seller (European call option) Let f(t) be the value of the option at t, 0 < t ≤ T. If we were already at time T (now we know S(T) ), the fair price is just. Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer. If, on the other hand, if S(T - dt) < X, she need do nothing. These two possibilities are described by: She should be holding shares.

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.)

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.) Formally: f depends on S, which varies randomly in time according to

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.) Formally: f depends on S, which varies randomly in time according to Use Ito’s lemma to see how f varies:

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.) Formally: f depends on S, which varies randomly in time according to Use Ito’s lemma to see how f varies:

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.) Formally: f depends on S, which varies randomly in time according to Use Ito’s lemma to see how f varies: The seller’s portfolio is

hedging strategy (continued) Farther back in time: Always hold shares. If S changes, buy shares. (Sell if.) Formally: f depends on S, which varies randomly in time according to Use Ito’s lemma to see how f varies: The seller’s portfolio is so it varies according to

deriving the Black-Scholes equation

Because of the hedging strategy, the noise has cancelled out!

deriving the Black-Scholes equation Because of the hedging strategy, the noise has cancelled out! Therefore the risk has been eliminated.

deriving the Black-Scholes equation Because of the hedging strategy, the noise has cancelled out! Therefore the risk has been eliminated. And because of that (as in the case of futures), the portfolio has to earn the T-bill rate r 0 :

deriving the Black-Scholes equation Because of the hedging strategy, the noise has cancelled out! Therefore the risk has been eliminated. And because of that (as in the case of futures), the portfolio has to earn the T-bill rate r 0 :

Black-Scholes equation Combining the two expressions for dΠ,

Black-Scholes equation Combining the two expressions for dΠ,

Black-Scholes equation Combining the two expressions for dΠ, Black-Scholes equation

Combining the two expressions for dΠ, Black-Scholes equation This looks “Fokker-Planck-ish”.

Black-Scholes equation Combining the two expressions for dΠ, Black-Scholes equation This looks “Fokker-Planck-ish”. It is in fact (except for the r 0 f term on the right-had side) and a flip of the time direction, the adjoint (backward) FP equation (or backward Kolmogorov equation).

Black-Scholes equation Combining the two expressions for dΠ, Black-Scholes equation This looks “Fokker-Planck-ish”. It is in fact (except for the r 0 f term on the right-had side) and a flip of the time direction, the adjoint (backward) FP equation (or backward Kolmogorov equation). (The S ’s occur in front of the derivative operators rather than after them.)

a digression on adjoint equations Recall for a Markov process

a digression on adjoint equations Recall for a Markov process What about calculating P n (t -1), given P n (t) ?

a digression on adjoint equations Recall for a Markov process What about calculating P n (t -1), given P n (t) ? governed by the transpose (adjoint) matrix T n’n:

a digression on adjoint equations Recall for a Markov process What about calculating P n (t -1), given P n (t) ? governed by the transpose (adjoint) matrix T n’n: For master equation, we had the forward equation

a digression on adjoint equations Recall for a Markov process What about calculating P n (t -1), given P n (t) ? governed by the transpose (adjoint) matrix T n’n: For master equation, we had the forward equation => backward equation is

some facts about adjoint matrices Suppose

some facts about adjoint matrices Suppose Then

some facts about adjoint matrices Suppose Then For differential operators, fx

adjoint Fokker-Planck operator FP operator:

adjoint Fokker-Planck operator FP operator: backward FP operator:

adjoint Fokker-Planck operator FP operator: backward FP operator: The solution of the FP equation

adjoint Fokker-Planck operator FP operator: backward FP operator: The solution of the FP equation The solution of the adjoint (or backward) FP equation

adjoint Fokker-Planck operator FP operator: backward FP operator: The solution of the FP equation The solution of the adjoint (or backward) FP equation describes the change of the probability density with the final time

back to Black-Scholes The BS equation:

back to Black-Scholes The BS equation: Boundary condition:

back to Black-Scholes First, get rid of the r 0 f term by defining The BS equation: Boundary condition:

back to Black-Scholes First, get rid of the r 0 f term by defining The BS equation: Boundary condition:

back to Black-Scholes First, get rid of the r 0 f term by defining The BS equation: Boundary condition: The boundary condition is now

solving Black-Scholes Define y = log S. Then

solving Black-Scholes Define y = log S. Then

solving Black-Scholes Define y = log S. Then

solving Black-Scholes Define y = log S. Then Except for the sign of t, this is a standard FP equation with constant drift and diffusion constants, so define τ = T – t, h(τ) = g(T - τ)

solving Black-Scholes Define y = log S. Then Except for the sign of t, this is a standard FP equation with constant drift and diffusion constants, so define τ = T – t, h(τ) = g(T - τ)

solving Black-Scholes Define y = log S. Then Except for the sign of t, this is a standard FP equation with constant drift and diffusion constants, so define τ = T – t, h(τ) = g(T - τ) with boundary condition

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be note sign 

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be For general h(y,0), note sign 

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be For general h(y,0), Here note sign 

solving Black-Scholes (2) Green’s function: If the initial condition were the solution would be For general h(y,0), Here note sign 

doing the integrals:

doing the integrals (2):

result: So the price of the European call option is

result: So the price of the European call option is

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1 effectively a future

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1 effectively a future in the limit σ -> 0

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1 effectively a future in the limit σ -> 0

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1 effectively a future in the limit σ -> 0 “at the money” ( X = S ), short maturity (small T ):

result: So the price of the European call option is in the limit S -> ∞, H( ) -> 1 effectively a future in the limit σ -> 0 “at the money” ( X = S ), short maturity (small T ):